HW2

Due Friday 09/08/2017 11:55:00 PM

These should help review content of Tuesday’s Lecture on Normal Theory. Please try to complete before class on Thursday in case there are questions or clarifications needed. Use LaTeX or write by hand (must be legible) and scan to submit via Sakai.

1. Exercise 1.11 Christensen (review notes on expectations and covariance; recall that $E[A Y] = A Cov$(A Y,B Y) = A$Cov$(Y)B^T$

2. Suppose $\Sigma$ is a real $p \times p$ positive semi-definite matrix. Then the Cholesky decomposition of $\Sigma = L L^T$ where $L$ is a lower triangular matrix with real, non-negative elements on the diagonal. Suppose you can generate standard normal random variates, $Z = (z_1, \ldots, z_p)^T$ with $z_i \sim N(0,1)$ (iid), then what is the distribution of $Y = \mu + L Z$ for $\mu \in \mathbb{R}^p$? Explain why it does not matter for generating $Y$ that the Cholesky decomposition is not unique when $\Sigma$ is not positive definite. Does it matter how we choose a matrix square root $A$ of $\Sigma = A A^T$? Explain.

3. Exercise 1.4 multivariate normal density (complete sketch from class notes)

4. Exercise 1.5.3

Review Chapter 1 in Plane Answers to Complex Questions on Vector Spaces

Review Material from the StatSci Bootcamp